Hermitian algebraic $K$-theory and the root system $D$

For the root system $D$, we construct an analog of the Wagoner complex used in his proof of the equivalence of $K^Q_*$ and $K^{BN}_*$ (linear) algebraic $K$-theories. We prove that the corresponding $K$-theory $KU^D_*$ for the even orthogonal group is naturally isomorphic to the $KU^{BN}_*$-theory constructed by Yu. P. Solovyov and A. I. Nemytov.